In radar imaging, a target is usually consisted of a few strong scatterers which are sparsely distributed. In this paper, an improved sparse signal recovery algorithm based on smoothed
l
_{0}
(SL0) norm method is proposed to achieve high resolution ISAR imaging with limited pulse numbers. Firstly, one new smoothed function is proposed to approximate the
l
_{0}
norm to measure the sparsity. Then a single loop step is used instead of two loop layers in SL0 method which increases the searching density of variable parameter to ensure the recovery accuracy without increasing computation amount, the cost function is undated in every loop for the next loop until the termination is satisfied. Finally, the new set of solution is projected into the feasible set. Simulation results show that the proposed algorithm is superior to the several popular methods both in terms of the reconstruction performance and computation time. Real data ISAR imaging obtained by the proposed algorithm is competitive to several other methods.
1. Introduction
I
nverse synthetic aperture radar (ISAR) imaging has received much attention in the past three decades
[1]

[3]
. ISAR imaging is widely used in many military and civilian applications, such as target identification and aircraft traffic control. In the conventional ISAR imaging, the observing interval must be long enough so that a high crossrange resolution can be obtained by a coherent integration. To obtain high resolution, ISAR imaging always needs many measurements in range frequency and crossrange time domains. For a long coherent processing interval, the target may moves with maneuvering. If the rotation angle is too large, the Radar Cross Section (RCS) of the scatterer may be time varying which increases the difficulty of coherent processing. So implementing imaging in a short time duration is meaningful.
In recent years, compressive sensing (CS) has become very popular in signal processing
[4]

[9]
. CS provides a new sampling paradigm which is able to reconstruct the sparse or compressible signals exactly from limited measurements by solving an optimization problem. It is a technique proposed to improve signal separation ability using a prior sparse property information of the signal. Sparsity usually can be measured by
l_{p}
(0 ≤
p
≤ 1) norm
[10]
. The sparse signal reovery is a key step in CS. Although
l
_{0}
norm is better in describing sparsity of noise free case, sparse signal recovery algorithms based on
l
_{0}
norm are intractable because they are sensitive to noise and need combinatorial search. The reconstruction algorithms based on
l
_{1}
norm are computational complex, which limits their practical applications. Hence many simpler algorithms, such as orthogonal matching pursuit (OMP)
[11]
[12]
are proposed. However, they are iteratively greedy algorithms and do not give good estimation of the sources.
Mohimani et al proposed a smoothed function to approximate
l
_{0}
norm, then the problem of minimum
l
_{0}
norm optimization can be transferred to an optimization problem of smoothed functions. The method called smoothed
l
_{0}
norm (SL0) method
[13]
. The SL0 method is about two orders of magnitude faster than
l
_{1}
magic method, while providing better estimation of the source than
l
_{1}
magic method.
For radar imaging, a target is usually regarded as consist of a few strong scatterers and the distribution of these strong scatterers is sparse in the imaging volume. Then sparse learning methods can be used to improve radar imaging quality
[14]

[17]
, such as SAR/ISAR imaging
[18]
[19]
, MIMO radar imaging
[20]
[21]
, and so on.
For ISAR imaging, this has been shown that combining ISAR and sparse learning can improve the 2D image quality with limited measured data
[22]
. Combination of local sparsity constraint and nonlocal total variation are discussed in
[23]
. The application of CS to ISAR imaging of moving targets in sea clutter is discussed in
[24]
. A multitask Bayesian model is utilized for ISAR in
[25]
. Fully polarimetric ISAR imaging based on CS is discussed in
[26]
.
In this paper, we proposed a new reconstruction algorithm based on SL0 method to improve ISAR imaging quality. One new continuous sequence is proposed as smoothed function to approach the
l
_{0}
norm which is suit to measure sparsity. Then one single loop step is used to replace two loop layers in SL0 algorithm which increase the searching density of variable parameters. The proposed algorithm ensures the reconstruction accuracy and the computation amounts don’t increase. By using the improved algorithm, ISAR imaging is more intensive with limited pulse numbers. Real data ISAR images obtained using the proposed method is competitive to the several popular methods.
This paper is organized as follows. Section 2 introduces necessary ISAR model and sparse learning imaging formation. In section 3, the proposed reconstruction algorithm is introduced in detail. Simulation and real data ISAR imaging results are presented in section 4. Finally, section 5 provides the conclusion and discussion.
2. ISAR Imaging Model Based On Sparse Learning
In order to facilitate analysis, it can be assumed thatthe translational motion of the target has been completely compensated via conventional methods. During the coherent processing interval (CPI), the radar transmits linear frequency modulation modulated signal can be defined as
where
is the fast time,
t
is the slow time, Δ
t
is the pulse repetition duration,
f_{c}
is the carrier frequency,
γ
is the chirp rate,
T_{P}
is the pulse width,
rect
(⋅) is the rectangle pulse function. Then the complex echo signal is
where
c
is the light speech,
T_{a}
is the CPI,
A
is the backward scattering amplitude which can be viewed stationary during the CPI. After the range compression, the received signal can be described as
where
λ
is the wavelength. At different dwell time
t
, the received signal has different time delay in the fast time
After pulse compression by matched filtering and omitting the constant introduced, the received signal becomes
where
f
= 2
xω
/
λ, β
= 2
xα
/
λ
are the Doppler and Doppler rate respectivel. Assuming a distance unit includes
K
scatterer points, the signal in the range cell corresponding to
τ
= 2(
R
_{0}
+
y
)/
c
by omitting the constant phase term can be written as
where
A_{k}
and
f_{k}
are the
k
th scattering centers’ reflecting amplitude and Doppler frequency, respectively.
n
is the additive noise. The time sequence is
t
= [1 :
N
]
^{T}
⋅Δ
t
, Δ
t
= 1/
f_{r}
, being the time interval,
f_{k}
is the pulse repetition frequency.
N
=
T
/Δ
t
is the number of pulses. Δ
f_{d}
is the Doppler frequency resolution, the sparse Doppler sequence is
f_{d}
= [1 :
Q
]⋅Δ
f_{d}
,
Q
=
f_{r}
/Δ
f_{d}
,
Q
is the number of Doppler unit corresponding to Δ
f_{d}
. Thus, the basis matrix can be constructed as Ψ = {
φ
_{1}
,
φ
_{2}
,⋯,
φ_{q}
,⋯,
φ_{Q}
},
φ_{q}
(
t
) = exp(
j
2
π
f_{d}
(
q
)
t
), 0 ≤
q
≤
Q
.
Then the received discrete signal equation can be rewritten as
The nonzero components of
θ
in the sparse vector correspond to the amplitudes of the strong scatterers which are located in the grids. For compressive sensing, the optimization algorithms have been applied for real number. Equation (6) should be transformed into real number case. We divide the signal into the real and imaginary components as follows
where ℜ (⋅) and ℑ (⋅) express the real and imaginary part of the complex vector respectively. So the equation (6) becomes
To solve
η
, we can use the following sparse optimization strategy
where
ε
is a small positive number associated with z.
p
indicates the
l_{p}
norm. The imaging quality largely depends on reconstruction algorithms, an improved SL0 algorithm is proposed and applied to ISAR imaging. We give a detailed description about the algorithm in the
3. The Improve SL0 Imaging Algorithm
To obtain an approximate
l
_{0}
norm solution, a smoothed function
was used to replace the
l
_{0}
norm in
[13]
. When a parameter
σ
approaches zero, the function
G_{σ}
(
η
) approaches
l
_{0}
norm. A twolayer method was proposed to solve the sparse signal recovery problem. In order to improve the approximation performance of the smoothed function, we propose a continuous sequence as follow
where
σ
is the variable parameter,
ς
is a small positive element which ensures the function is continuous and differentiable. It is obvious that
Then denote
where 
η

_{0}
expresses the number of nonzero elements of vector
η
. According to the definition of
l
_{0}
norm, we can obtain
A property of
G_{σ}
(
η
) is that when
σ
→ ∞,
N

G_{σ}
(
η
) approaches
l
_{2}
norm. However, for
F_{σ,ς}
(
η
) when
σ
→ ∞,
N

F_{σ,ς}
(
η
) approaches
l
_{1}
norm. When
σ
approximates zero, it approaches
l
_{0}
norm. For
l
_{1}
norm can describe sparsity, we can search for the sparse solution with high probability at the beginning of iteration.
Therefore sparse signal recovery algorithm based on function
F_{σ,ς}
(
η
) minimum can be described as
In the proposed algorithm, we use one loop layer to replace two loop layers in SL0 algorithm. For smoothed
l
_{0}
norm algorithm presented in
[13]
, two layer loop is used to obtain the minimum solution. Generally speaking, if the inner iterative number is large enough, the step size can be small. Actually, obtaining a precise solution in inner loop is not necessary. The aim of inner loop is to provide an initial value for the outer loop. Since the double loop does not need to find out the real point, we use single layer to replace two loop layers and increase the searching density of variable parameter
σ
at the same time which search a point approaches to the minimum solution in each iteration. The proposed algorithm called ISSL0. We add a step which compare the old cost function and new cost function. If Newfunction is greater than Oldfunction, the loop stops, otherwise, continues to the next loop. When the parameter
σ
approaches zero,
F_{σ}
approaches
l
_{0}
norm. If New function is greater than Old function, the solution is the minimum point at this time. Then the loop stops which can save the computation amount. The ISSL0 algorithm ensures the reconstruction accuracy and the computation amount don’t increase compared with SL0 algorithm. The gradient projection method is used to project new iteration position to the feasible set. The total ISSL0 optimization algorithm in this paper can be summarized as follows:
Initialization:
1) Let
be equal to the minimum
l
_{2}
norm solution of
y
=
Aη
, obtained by
2) Choose a suitable decreasing sequence for {
σ
}, [
σ
_{1}
,⋯,
σ_{J}
].

forj= 1,⋯,J

(1) Letσ=σj,

Oldfunction =Nsum(Fσ,ς(η)).

(2) Minimize the functionFσ,ς(η) on the feasible setη= {η:Aηy2<ε}

a) Initialization:

b) Letδbe gradient ofFσ,ς(η).

c) For every element ofη, letη(n) ←η(n) μσδ(n) whereμis a small positive constant)

d) If Aηy2>ε, projectηback into the feasible setη:η←ηAH(AAH)1(ηAη)

e) updateFσ,ς(η).
3) Set
4) Newfunction =
N

sum
(
F_{σ,ς}
(
η
))

If Newfunction>Oldfunction

break

end
Final answer is
In the above algorithm, some initial parameters should be chosen.
is the minimum
l
_{2}
norm solution. In
[13]
,
σ
_{1}
is chosen as
In this paper,
σ
_{1}
should be chosen as
For
l
_{0}
norm is not suited to express a vector with many small elements, the choice of
σ_{J}
should not be too small.
σ_{J}
can be estimated by selecting a few noise samples, selecting the maximum value of
and taking the average value. We choose
σ_{J}
For the choice of step size factor
μ
, at the beginning of search, we select a larger step size, when the searching point approaches the minimum solution, the step size should decrease. So in our algorithm, we choose step size as
μ
=
β
max
η
/10 with the increase of
j
(loop number), the adjust factor
β
decreases, then the step size decreases. max
η
 term adjusts the step size to match the solution. The size of is
M
×
N
. The computational loads of the proposed algorithm are consisted of the step (c) to (d) in step 2) of each iteration.
Aη
needs
MN
multiplications in each iteration. So the computational complexity of the proposed method is
O
(
MN
).
4. Experimental Results
 Simulation 1. One Dimensional Synthetic Signals Recovery
The signal model with noise is
y
=
Aη
+
z
, the size of A (
M
×
N
) is 128×256, it is constructed by selecting its components from
N
(0,1).
η
is the sparse signal, whose nonzero coefficients are uniform 1± random spikes signal. We consider SNR=15, 20,25,30dB conditions. For SL
_{0}
method, the numbers of outer loop and inner loop are 20 and 10 respectively. For ISSL0 algorithm, the loop number is 200. The parameter
ς
= 0.01. The MSE is defined as
where
η
is the true solution and
is the estimation value. The experiment was implemented 100 times (with the same parameters, but for different randomly generated sources and coefficient matrices). The computation time , reconstruction probability and MSE are averaged.
Fig. 1
shows the average computational time for 15dB SNR case. For other SNR cases, the computational time is similar to this case. From
Fig. 1
, we can see the computation costs of OMP , SL0 and ISSL0 are less than Bayesian and l1ls methods.The reconstruction probability and MSE of different methods with different
K
are shown in
Fig. 2
and
Fig. 3
. The reconstruction probability of different methods decreases gradually with K increases. MSE of different methods increases gradually with K increases. We can see that with the increase of SNR, the performance of l1ls and ISSL0 methods improve faster than OMP, SL0 and Bayesian methods. The performances of ISSL0 algorithm are competitive with other algorithms especially when SNR is high and K is large.
Computation costs of different methods
Correct position estimation for different K (a) SNR=15dB (b)SNR=20dB (c) SNR =25dB, (d) SNR = 30dB
MSE for different K (a) SNR = 15dB, (b) SNR = 20 dB (c) SNR =25dB, (d) SNR = 30 dB
 Simulation 2. ISAR imaging using real data
In this section, a set of real data of the Yak42 plane is used to demonstrate the performance of the proposed ISAR imaging algorithm. The related parameters descriptions of the radar data are listed as follows: the carrier frequency is 10 GHz with signal bandwidth of 400 MHz, a range resolution is 0.375 m. The pulse repetition frequency is 50 Hz, i.e., 256 pulses are used in this experiment. Two different amounts of pulses (32snapshot and 64snapshot) are implemented. The ISAR images are reconstructed with 256 Doppler bins (that means the size of ISAR is 256 × 256) is shown as
Fig. 4
(a). The experimental results are compared visually and quantitatively to those images obtained by some sparse signal recovery methods including OMP, Bayesian method with Laplace prior and SL0 methods. From
Fig. 4
(b), (c), (d), (e),(f) and
Fig. 5
. It is noticeable that more amount of pulses generally lead to better imagery results. For 32 snapshots case, images using OMP method has many clutter points. Compared with the other four imaging methods, the proposed ISAR imaging framework generates a better visual quality and has competitive performance.We can see that the ISSL0 method generates a better visual quality and the ISAR imaging is more intensive. The noticeable advantage of the ISSL0 imaging method is that the strong scatterers of the target are extracted well along with fewer false points. The time of 32 snapshots is only one eighth of the original 256 snapshots acquirement time. It is possible to image maneuvering target in a short time duration using sparse signal recovery algorithms.
(a) ISAR image using 256 samples by FFT method, Reconstructed images using 32 snapshots, (b) OMP (c) Bayesian method (d) l1ls (e) SL0 (f) ISSL0
Reconstructed images using 64 snapshots, (a) OMP (b) Bayesian method (c) l1ls (d)SL0 (e) ISSL0
5. Conclusion
In this paper, one improved sparse signal recovery algorithm based on SL0 algorithm is demonstrated to improve ISAR imaging with limited pulse numbers. We propose one new continuous function as smoothed function sequence to approximate
l
_{0}
norm, then single loop step is used to replace two loop layers in SL0 algorithm to ensure the reconstruction accuracy without increasing computation amounts. Simulation results show that the performance of ISSL0 algorithm is superior than OMP, Bayesian method with Laplace prior, l1ls and SL0 methods. The real data experiments show the proposed algorithm can improve imaging quality.
BIO
Junjie Feng received the B.E. and M.S. degrees from the Baoji University of Arts and Science and the Zhongyuan University of Technology in 2006 and 2010, respectively. He is currently pursuing the Ph.D. degree with the Nanjing University of Aeronautics and Astronautics, China. His research interests include radar signal processing and wireless communication.
Gong Zhang received the Ph.D. degree in electronic engineering from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2002. From 1990 to 1998, he was a Member of Technical Staff at No724 Institute China Shipbuilding Industry Corporation (CSIC), Nanjing. Since 1998, he has been with the College of Electronic and Information Engineering, NUAA, where he is currently a Professor. His research interests include radar signal processing and classification recognition. Dr. Zhang is a member of Committee of Electromagnetic Information, Chinese Society of Astronautics (CEICSA) and a Senior Member of the Chinese Institute of Electronics (CIE).
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